Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 311 / 1 / a

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- Solution a

Solution

Every metric component depends only on

r

, so translations of

t

and

ϕ

leave the metric unchanged. Equivalently,

L_{\partial_{t}} g_{ab} = \partial_{t} g_{ab} = 0, L_{\partial_{ϕ}} g_{ab} = \partial_{ϕ} g_{ab} = 0.

Thus

k = \partial_{t}

and

m = \partial_{ϕ}

are Killing vector fields, generating stationarity and axial symmetry respectively.

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